How Much Work Is Needed to Push a Crate Up an Inclined Plane?

[mstine87]

I am a stagehand and I was just doing a job yesterday involving running crates on wheels up a ramp to the loading bay. I thought it would be interesting if I could make a physics equation out of it and determine the amount of work I did.

Homework Statement

If I want to determine how much mechanical work (J) is done by accelerating a crate on wheels with a mass (m) of 90kg, and pushing it at a constant velocity (v) of 2m/s up a plane inclined at 15 degrees (It's on wheels, so effectively frictionless plane). The ramp is 30 m in distance, and total vertical displacement (h) would be approximately 10.26 m.

I determined that I was reaching the top of the ramp in about 15 seconds, which is how I determined that my average velocity was 2m/s.[/b]

Homework Equations

1. v=Δd/Δt
2. a=Δv/Δt
3. f=m*a
4. w=f*d
5. g=9.8m/s^2

Derivative equation
6. a=g*sin(theta)
7.f=m*g*cos(theta)

The Attempt at a Solution

The Force of gravity on the 90 kg mass is given by (eq3), or f=90 kg*9.8m/s^2. f(g)=882N

I figured that I could determine the force of the mass pushing down the slope against me by determining its rate of acceleration down the slope if I weren't there
(eq6) a=9.8*sin(15 degrees), a=2.5 m/s^2
The force down the slope is then (eq3)f=90kg*2.5m/s^2=225N.
I think this means that I would have to apply a force of 225N to keep the crate still on the plane- precisely opposing the force of gravity pulling it down.

The normal force is given by (eq7). m*g was calculated earlier as 882N, and 882*cos(15 degrees) is 851.94N.

Here is my problem. I can't figure out what I need to do next. I want to know how much work was done to get the mass to the top of the slope. So I would need to know how much force I needed to push the crate at a constant velocity of 2m/s, and I have no idea how to get there. Once I know the total force I applied to overcome the force of gravity, I can add in the force it would take to accelerate the mass to 2m/s and keep it there. Can anybody help me?

I would like to add that this is not homework or school related. I am interested in applying what I learned when I was in school to my job for leisure.

 

[ehild]

Neglecting all kinds of friction, you need the same force to push the crate uphill with constant velocity as to keep it in rest: mgsin(theta).
When you accelerate the cart, you need an additional force ma.

You can solve the problem in this way, multiplying force with displacement, but it is easier to use energy: Your work increases both the kinetic and potential energy of the crate. Calculate the energy of the crate at the top of the slope: That is your work done on it while it is accelerated and pushed to the top of the slope.

ehild